Properties

Label 475.128
Modulus $475$
Conductor $475$
Order $180$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([63,70]))
 
pari: [g,chi] = znchar(Mod(128,475))
 

Basic properties

Modulus: \(475\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 475.bi

\(\chi_{475}(2,\cdot)\) \(\chi_{475}(3,\cdot)\) \(\chi_{475}(13,\cdot)\) \(\chi_{475}(22,\cdot)\) \(\chi_{475}(33,\cdot)\) \(\chi_{475}(48,\cdot)\) \(\chi_{475}(52,\cdot)\) \(\chi_{475}(53,\cdot)\) \(\chi_{475}(67,\cdot)\) \(\chi_{475}(72,\cdot)\) \(\chi_{475}(78,\cdot)\) \(\chi_{475}(97,\cdot)\) \(\chi_{475}(98,\cdot)\) \(\chi_{475}(108,\cdot)\) \(\chi_{475}(117,\cdot)\) \(\chi_{475}(127,\cdot)\) \(\chi_{475}(128,\cdot)\) \(\chi_{475}(147,\cdot)\) \(\chi_{475}(148,\cdot)\) \(\chi_{475}(162,\cdot)\) \(\chi_{475}(167,\cdot)\) \(\chi_{475}(173,\cdot)\) \(\chi_{475}(192,\cdot)\) \(\chi_{475}(203,\cdot)\) \(\chi_{475}(212,\cdot)\) \(\chi_{475}(222,\cdot)\) \(\chi_{475}(223,\cdot)\) \(\chi_{475}(238,\cdot)\) \(\chi_{475}(242,\cdot)\) \(\chi_{475}(262,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((77,401)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{7}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 475 }(128, a) \) \(1\)\(1\)\(e\left(\frac{133}{180}\right)\)\(e\left(\frac{91}{180}\right)\)\(e\left(\frac{43}{90}\right)\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{1}{90}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{107}{180}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 475 }(128,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 475 }(128,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 475 }(128,·),\chi_{ 475 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 475 }(128,·)) \;\) at \(\; a,b = \) e.g. 1,2